Weak convergence and empirical processes: with applications to statistics
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2000
|
| Edition: | Corr. 2. print. |
| Series: | Springer series in statistics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 508 S. |
| ISBN: | 0387946403 9781475725476 |
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MARC
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| 100 | 1 | |a Vaart, Aad W. van der |d 1959- |e Verfasser |0 (DE-588)114474680 |4 aut | |
| 245 | 1 | 0 | |a Weak convergence and empirical processes |b with applications to statistics |c Aad W. van der Vaart ; Jon A. Wellner |
| 250 | |a Corr. 2. print. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 2000 | |
| 300 | |a XVI, 508 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer series in statistics | |
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| 700 | 1 | |a Wellner, Jon A. |d 1945- |e Verfasser |0 (DE-588)170864847 |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-1-4757-2545-2 |
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Record in the Search Index
| _version_ | 1804128457592930304 |
|---|---|
| adam_text | Contents
Preface
..........................
vii
Reading
Guide
...................... xi
1.
Stochastic Convergence
................1
1.1.
Introduction
.......................2
1.2.
Outer Integrals and Measurable
Majorants
..........6
1.3.
Weak Convergence
...................16
1.4.
Product Spaces
.....................29
1.5.
Spaces of Bounded Functions
...............34
1.6.
Spaces of Locally Bounded Functions
...........43
1.7.
The Ball
Sigma-Field
and Measurability of
Suprema
.....45
1.8.
Hubert Spaces
.....................49
1.9.
Convergence: Almost Surely and in Probability
.......52
1.10.
Convergence: Weak, Almost Uniform, and in Probability
... 57
1.11.
Refinements
...................... 67
1.12.
Uniformity and Metrization
................ 71
Notes
......................... 75
2.
Empirical Processes
.................79
2.1.
Introduction
......................80
2.1.1.
Overview of Chapters
2.3-2.14........... 83
2.1.2.
Asymptotic Equicontinuity
............ 89
2.1.3.
Maximal Inequalities
............... 90
*2.1.4. The Central Limit Theorem in Banach Spaces
.... 91
2.2.
Maximal Inequalities and Covering Numbers
........ 95
2.2.1.
Sub-Gaussian Inequalities
............. 100
2.2.2.
Bernstein s Inequality
.............. 102
*2.2.3. Tightness Under an Increment Bound
....... 104
2.3.
Symmetrization and Measurability
............. 107
2.3.1.
Symmetrization
.................107
*2.3.2. More Symmetrization
...............
Ill
*2.3.3. Separable Versions
................115
2.4.
Glivenko-Cantelli Theorems
................122
2.5.
Donsker Theorems
....................127
2.5.1.
Uniform Entropy
.................127
2.5.2.
Bracketing
...................129
2.6.
Uniform Entropy Numbers
................134
2.6.1.
VC-Classes of Sets
................ 134
2.6.2.
VC-Classes of Functions
............. 140
2.6.3.
Convex Hulls and VC-Hull Classes
......... 142
2.6.4.
VC-Major Classes
................ 145
2.6.5.
Examples and Permanence Properties
....... 146
2.7.
Bracketing Numbers
................... 154
2.7.1.
Smooth Functions and Sets
............154
2.7.2.
Monotone Functions
...............159
2.7.3.
Closed Convex Sets and Convex Functions
.....162
2.7.4.
Classes That Are Lipschitz in a Parameter
.....164
2.8.
Uniformity in the Underlying Distribution
.........166
2.8.1.
Glivenko-Cantelli Theorems
............166
2.8.2.
Donsker Theorems
................168
2.8.3.
Central Limit Theorem Under Sequences
......173
2.9.
Multiplier Central Limit Theorems
............176
2.10.
Permanence of the Donsker Property
...........190
2.10.1.
Closures and Convex Hulls
............190
2.10.2.
Lipschitz Transformations
.............192
2.10.3.
Permanence of the Uniform Entropy Bound
.....198
2.10.4.
Partitions of the Sample Space
..........200
2.11.
The Central Limit Theorem for Processes
.........205
2.11.1.
Random Entropy
................205
2.11.2.
Bracketing
...................210
2.11.3.
Classes of Functions Changing with
η
.......220
2.12.
Partial-Sum Processes
..................225
2.12.1.
The Sequential Empirical Process
.........225
2.12.2.
Partial-Sum Processes on Lattices
.........228
2.13.
Other Donsker Classes
..................232
2.13.1.
Sequences
....................232
2.13.2.
Elliptical Classes
.................233
2.13.3.
Classes of Sets
..................236
2.14.
Tail Bounds
......................238
2.14.1.
Finite Entropy Integrals
.............238
2.14.2.
Uniformly Bounded Classes
............245
2.14.3.
Deviations from the Mean
.............254
2.14.4.
Proof of Theorem
2.14.13.............257
Notes
.........................269
3.
Statistical Applications
...............277
3.1.
Introduction
......................278
3.2.
M-Estimators
......................284
3.2.1.
The Argmax Theorem
..............285
3.2.2.
Rate of Convergence
...............289
3.2.3.
Examples
....................294
3.2.4.
Linearization
..................300
3.3.
Z-Estimators
......................309
3.4.
Rates of Convergence
.................. 321
3.4.1.
Maximum Likelihood
...............326
3.4.2.
Concave Parametrizations
............. 330
3.4.3.
Least Squares Regression
.............331
3.4.4.
Least-Absolute-Deviation Regression
........336
3.5.
Random Sample Size, Poissonization and
Кас
Processes
. . . 339
3.5.1.
Random Sample Size
...............339
3.5.2.
Poissonization
..................341
3.6.
The Bootstrap
.....................345
3.6.1.
The Empirical Bootstrap
.............345
3.6.2.
The Exchangeable Bootstrap
...........353
3.7.
The Two-Sample Problem
................360
3.7.1.
Permutation Empirical Processes
..........362
3.7.2.
Two-Sample Bootstrap
..............365
3.8.
Independence Empirical Processes
.............367
3.9.
The Delta-Method
....................372
3.9.1.
Main Result
...................372
3.9.2.
Gaussian Limits
.................376
3.9.3.
The Delta-Method for the Bootstrap
........377
3.9.4.
Examples of the Delta-Method
...........381
3.10.
Contiguity
.......................401
3.10.1.
The Empirical Process
..............406
3.10.2.
Change-
Point
Alternatives
............408
3.11.
Convolution
and
Minimax
Theorems
............412
3.11.1.
Efficiency of the Empirical Distribution
.......420
Notes
.........................423
A. Appendix
.......................429
A.I. Inequalities
.......................430
A.
2.
Gaussian Processes
...................437
A.
2.1.
Inequalities and Gaussian Comparison
.......437
A.2.2. Exponential Bounds
...............442
A.2.3. Majorizing Measures
...............445
A.2.4. Further Results
.................447
A.3. Rademacher Processes
..................449
A.4. Isoperimetric Inequalities for Product Measures
.......451
A.
5.
Some Limit Theorems
..................456
A.
6.
More Inequalities
....................459
A.6.1. Binomial Random Variables
............459
A.6.2. Multinomial Random Vectors
...........462
A.6.3. Rademacher Sums
................463
Notes
.........................465
References
........................467
Author Index
......................487
Subject Index
......................493
List of Symbols
.....................506
|
| any_adam_object | 1 |
| author | Vaart, Aad W. van der 1959- Wellner, Jon A. 1945- |
| author_GND | (DE-588)114474680 (DE-588)170864847 |
| author_facet | Vaart, Aad W. van der 1959- Wellner, Jon A. 1945- |
| author_role | aut aut |
| author_sort | Vaart, Aad W. van der 1959- |
| author_variant | a w v d v awvd awvdv j a w ja jaw |
| building | Verbundindex |
| bvnumber | BV013648340 |
| classification_rvk | QH 237 SK 820 |
| classification_tum | MAT 604f |
| ctrlnum | (OCoLC)248133431 (DE-599)BVBBV013648340 |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | Corr. 2. print. |
| format | Book |
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| id | DE-604.BV013648340 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:49:33Z |
| institution | BVB |
| isbn | 0387946403 9781475725476 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009324519 |
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| publishDateSearch | 2000 |
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| publisher | Springer |
| record_format | marc |
| series2 | Springer series in statistics |
| spelling | Vaart, Aad W. van der 1959- Verfasser (DE-588)114474680 aut Weak convergence and empirical processes with applications to statistics Aad W. van der Vaart ; Jon A. Wellner Corr. 2. print. New York [u.a.] Springer 2000 XVI, 508 S. txt rdacontent n rdamedia nc rdacarrier Springer series in statistics Empirischer Prozess - Stochastische Konvergenz Empirischer Prozess (DE-588)4224810-3 gnd rswk-swf Stochastische Konvergenz (DE-588)4183376-4 gnd rswk-swf Empirischer Prozess (DE-588)4224810-3 s Stochastische Konvergenz (DE-588)4183376-4 s DE-604 Wellner, Jon A. 1945- Verfasser (DE-588)170864847 aut Erscheint auch als Online-Ausgabe 978-1-4757-2545-2 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009324519&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Vaart, Aad W. van der 1959- Wellner, Jon A. 1945- Weak convergence and empirical processes with applications to statistics Empirischer Prozess - Stochastische Konvergenz Empirischer Prozess (DE-588)4224810-3 gnd Stochastische Konvergenz (DE-588)4183376-4 gnd |
| subject_GND | (DE-588)4224810-3 (DE-588)4183376-4 |
| title | Weak convergence and empirical processes with applications to statistics |
| title_auth | Weak convergence and empirical processes with applications to statistics |
| title_exact_search | Weak convergence and empirical processes with applications to statistics |
| title_full | Weak convergence and empirical processes with applications to statistics Aad W. van der Vaart ; Jon A. Wellner |
| title_fullStr | Weak convergence and empirical processes with applications to statistics Aad W. van der Vaart ; Jon A. Wellner |
| title_full_unstemmed | Weak convergence and empirical processes with applications to statistics Aad W. van der Vaart ; Jon A. Wellner |
| title_short | Weak convergence and empirical processes |
| title_sort | weak convergence and empirical processes with applications to statistics |
| title_sub | with applications to statistics |
| topic | Empirischer Prozess - Stochastische Konvergenz Empirischer Prozess (DE-588)4224810-3 gnd Stochastische Konvergenz (DE-588)4183376-4 gnd |
| topic_facet | Empirischer Prozess - Stochastische Konvergenz Empirischer Prozess Stochastische Konvergenz |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009324519&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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